Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without using exponents. $$ \left(a b^{2}\right)^{0}, \text { where } a, b \neq 0 $$

Step-by-Step Solution

Answer

\(\left(ab^2\right)^0 = 1\)

Step 1: Apply exponent rule to rewrite the expression

We know that any base raised to the power of 0 is equal to 1. Here, the entire expression inside the parentheses is raised to the power of 0, so we can rewrite it as: $$ \left(ab^2\right)^0 = 1 $$

Step 2: Simplify the expression

Since the entire expression has been simplified to equal 1, there are no more exponents and we have reached the final simplified form: $$ \left(ab^2\right)^0 = 1 $$

Key Concepts

Simplifying Exponents

Exponents can often appear daunting at first glance, especially when dealing with complex expressions. However, once you understand the rules that govern them, simplifying exponents becomes much more manageable.

When simplifying, it helps to break down the expression into individual parts and apply the relevant exponent rules. For example, when multiplying like bases, you add their exponents. Conversely, when dividing like bases, you subtract the exponents. It's crucial to remember that these processes are reversible, aiding in simplification.

Product of Powers

For instance, when you encounter an expression like \(x^m \times x^n\), you can simplify it to \(x^{m+n}\). Likewise, \(x^{m} \/ x^{n}\) simplifies to \(x^{m-n}\).

Power of a Power

Another key concept is when a power is raised to another power, such as \( (x^m)^n \), you multiply the exponents to simplify, resulting in \(x^{mn}\).

By understanding these rules and applying them step-by-step, you can greatly simplify expressions with exponents.

Zero Exponent Rule

The zero exponent rule is a fundamental concept that often simplifies seemingly complex expressions into a much simpler form. This rule states that any non-zero base raised to the power of zero equals one, symbolically \(a^0 = 1\), provided that \(a \eq 0\).

This rule is essential because it eliminates the need for further calculation; once you identify a term with a zero exponent, you know the value of that term is 1. This simplification step can significantly reduce the complexity of an equation or expression.

In the given exercise, \(\left(a b^{2}\right)^{0}\), the rule has been applied to simplify the entire expression to 1, regardless of the values of \(a\) or \(b\), as long as they are not zero. This showcases the power of the zero exponent rule in simplifying expressions.

Mathematical Expressions

Mathematical expressions consist of numbers, variables, and operations that, when combined, form a statement of a value or relationship. They are the foundation of algebra and essential for solving equations.

Breaking down expressions into elementary parts and understanding the role of each—whether it's a coefficient, variable, exponent, or constant—is key to simplifying and solving them. Expressions can be simplified by combining like terms, using distributive properties, and applying exponent rules.

Honing the skill to interpret and manipulate these expressions is vital for mathematics students. It involves recognizing patterns, applying laws correctly, and often simplifying complex information into a more manageable form to find a clear, succinct solution.